A basic task in quantum computing is to obtain information about the state of a quantum system by making measurements. If many identically prepared copies of a state are available, one can ask whether it is possible to uniquely determine the state from the statistical data produced by suitably chosen measurements. This problem of estimating an unknown quantum state is also referred to as quantum state tomography.
Quantum state tomography is perhaps one of the most fundamental problems in quantum information processing. At the same time, it provides one of the first potential physics applications for which a quantum computer might be useful. While the problem has been solved, at least conceptually, for the case of non-degenerate measurements of von Neumann type, the problem of quantum state tomography with highly degenerate measurements—a yes/no measurement being the extreme case of a degenerate measurement—has not been addressed thus far. The need for a solution for the case of yes/no measurements arises for instance in quantum computers built on the principles of nuclear magnetic resonance (NMR). In NMR systems it is typically not possible to measure all spins (qubits) of the system. A common assumption in NMR quantum computing is that there is only one pure qubit that can be measured after a quantum computation has been performed.
In principle, as has been described in the physics literature, a solution to the quantum state tomography problem is based on the understanding that knowledge of the statistics of a so-called informationally complete measurement is sufficient to infer the state of the system. There are several constructions for such measurements, with the leading approach being measurements in so-called mutually unbiased bases which correspond to non-commuting observables that optimally capture the features of the system.
The approaches considered so far in the literature, however, have one or more disadvantages. For instance, they are based on non-degenerate measurements, meaning that all of the system's qubits must be measured. While these measurements are convenient from a theoretical point of view, practically it is much more convenient if only one qubit is dedicated for readout, i.e., can be measured. Another limitation is that many informationally complete measurements are of the very general type of a positive operator valued measure (POVM). These, however, do not readily lend themselves to implementation on a quantum computer because one would have to translate the POVM into a quantum circuit first.
While quantum state tomography using yes/no measurements has been studied, so far only systems of dimensions of a power of two have been considered, which greatly restricts the possible systems in which state tomography can be performed.